#### 2.420   ODE No. 420

$a+x y'(x)^2-2 y(x) y'(x)=0$ Mathematica : cpu = 0.296321 (sec), leaf count = 777

$\left \{\left \{y(x)\to \frac {1}{4} a^2 e^{-\frac {3 c_1}{2}} x^2+\frac {1}{4} e^{-\frac {3 c_1}{2}} \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {-a^9 e^{3 c_1} x^9+3 a^6 e^{6 c_1} x^6-3 a^3 e^{9 c_1} x^3+e^{12 c_1}}-8 e^{6 c_1}}-\frac {e^{-\frac {3 c_1}{2}} \left (-9 a^4 x^4-72 a e^{3 c_1} x\right )}{36 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {-a^9 e^{3 c_1} x^9+3 a^6 e^{6 c_1} x^6-3 a^3 e^{9 c_1} x^3+e^{12 c_1}}-8 e^{6 c_1}}}\right \},\left \{y(x)\to \frac {1}{4} a^2 e^{-\frac {3 c_1}{2}} x^2-\frac {1}{8} \left (1-i \sqrt {3}\right ) e^{-\frac {3 c_1}{2}} \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {-a^9 e^{3 c_1} x^9+3 a^6 e^{6 c_1} x^6-3 a^3 e^{9 c_1} x^3+e^{12 c_1}}-8 e^{6 c_1}}+\frac {\left (1+i \sqrt {3}\right ) e^{-\frac {3 c_1}{2}} \left (-9 a^4 x^4-72 a e^{3 c_1} x\right )}{72 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {-a^9 e^{3 c_1} x^9+3 a^6 e^{6 c_1} x^6-3 a^3 e^{9 c_1} x^3+e^{12 c_1}}-8 e^{6 c_1}}}\right \},\left \{y(x)\to \frac {1}{4} a^2 e^{-\frac {3 c_1}{2}} x^2-\frac {1}{8} \left (1+i \sqrt {3}\right ) e^{-\frac {3 c_1}{2}} \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {-a^9 e^{3 c_1} x^9+3 a^6 e^{6 c_1} x^6-3 a^3 e^{9 c_1} x^3+e^{12 c_1}}-8 e^{6 c_1}}+\frac {\left (1-i \sqrt {3}\right ) e^{-\frac {3 c_1}{2}} \left (-9 a^4 x^4-72 a e^{3 c_1} x\right )}{72 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {-a^9 e^{3 c_1} x^9+3 a^6 e^{6 c_1} x^6-3 a^3 e^{9 c_1} x^3+e^{12 c_1}}-8 e^{6 c_1}}}\right \}\right \}$ Maple : cpu = 0.182 (sec), leaf count = 689

$\left \{ y \left ( x \right ) ={\frac {x}{12\,{\it \_C1}} \left ( 4\,{\frac {{x}^{2}}{\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}}}+2\,x+\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}} \right ) }+3\,{{\it \_C1}\,a \left ( 4\,{\frac {{x}^{2}}{\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}}}+2\,x+\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}} \right ) ^{-1}},y \left ( x \right ) =-{\frac {x}{6\,{\it \_C1}} \left ( {{x}^{2} \left ( 1+i\sqrt {3} \right ) {\frac {1}{\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}}}}-{\frac {i}{4}}\sqrt {3}\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}-x+{\frac {1}{4}\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}} \right ) }-6\,{{\it \_C1}\,a \left ( 4\,{\frac {{x}^{2} \left ( 1+i\sqrt {3} \right ) }{\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}}}-i\sqrt {3}\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}-4\,x+\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}} \right ) ^{-1}},y \left ( x \right ) ={\frac {x}{6\,{\it \_C1}} \left ( {{x}^{2} \left ( i\sqrt {3}-1 \right ) {\frac {1}{\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}}}}-{\frac {i}{4}}\sqrt {3}\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}+x-{\frac {1}{4}\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}} \right ) }+6\,{{\it \_C1}\,a \left ( 4\,{\frac {{x}^{2} \left ( i\sqrt {3}-1 \right ) }{\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}}}-i\sqrt {3}\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}}+4\,x-\sqrt [3]{-36\,a{{\it \_C1}}^{2}+8\,{x}^{3}+12\,\sqrt {a \left ( 9\,a{{\it \_C1}}^{2}-4\,{x}^{3} \right ) }{\it \_C1}} \right ) ^{-1}} \right \}$